The Stacks project

Lemma 74.42.8. Let $S$ be a scheme. Let $X$ be a Noetherian algebraic space over $S$ and let $\mathcal{I} \subset \mathcal{O}_ X$ be a quasi-coherent sheaf of ideals. Let $\mathcal{G}$ be a coherent $\mathcal{O}_ X$-module, $(\mathcal{F}_ n)$ an object of $\textit{Coh}(X, \mathcal{I})$, and $\alpha : (\mathcal{F}_ n) \to \mathcal{G}^\wedge $ a map whose kernel and cokernel are annihilated by a power of $\mathcal{I}$. Then there exists a unique (up to unique isomorphism) triple $(\mathcal{F}, a, \beta )$ where

  1. $\mathcal{F}$ is a coherent $\mathcal{O}_ X$-module,

  2. $a : \mathcal{F} \to \mathcal{G}$ is an $\mathcal{O}_ X$-module map whose kernel and cokernel are annihilated by a power of $\mathcal{I}$,

  3. $\beta : (\mathcal{F}_ n) \to \mathcal{F}^\wedge $ is an isomorphism, and

  4. $\alpha = a^\wedge \circ \beta $.

Proof. The uniqueness and étale descent for objects of $\textit{Coh}(X, \mathcal{I})$ and $\textit{Coh}(\mathcal{O}_ X)$ implies it suffices to construct $(\mathcal{F}, a, \beta )$ étale locally on $X$. Thus we reduce to the case of schemes which is Cohomology of Schemes, Lemma 30.23.6. $\square$


Comments (0)


Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.




In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 08BB. Beware of the difference between the letter 'O' and the digit '0'.